[ 837 ]
XXXIV. On the Construction of lAfe-Tahles^ illustrated by a New Life-Tdble of the
Healthy Districts of England, By W. Fare, Usq., M.B., F.EM,
Beceiyed March 17, — Bead April 7, 1859.
I. aEISTEEAL DESOEIPTION OF A LIFE-TAELE (Table C.) 839—841
{a) Age (^) ..,., ... 839
(5) Two primary series representing (1) the surviving (4) at each year of age, and (2) the
first differences of the same, representing the dying (4?) in annual intervals of age ...... 839 — 840
Plate XLII. figs. 1 and 2, representing by lines the series h and d^ 840 — 841
II. PEINCIPLES OF OOHSTEUCTION 841—849
(1) Hypothesis of an invariable annual number of births, equalling the deaths, has never
been observed in any city or co\intry ; and consequently Tables on the plan of Hallet' s
are often exceedingly erroneous, as in the case of the old JSForthampton Table 841
(2) The direct method of construction from the ascertained rates of mortality {m^ in
several periods of life 841 — 842
(3) (a) By Demoivee's hypothesis, the series Ix is an arithmetical progression 842
(5) The series h on the hypothesis that the cause of death is of equal intensity at all ages. 842 — 843
(c) The hypothesis that the values mx form geometrical progressions: rate of mor-
(d) Application of this hypothesis to the construction of Life-Tables by Mr. GtOMPebtz
and by Mr. Epmofds. Investigation of the formula. The limits to the applica-
tion of the hypothesis ,.. 844 — 848
{e) The determination of j?^; difference of ^^ and {l"-j?.f). Comparison of ^^ derived
from the formulae ( , . ^ - ), and ^=10^** 848
Vl+iw^/'
(/) The determination of the values oip^ at the earlier ages 848 — 849
III. INTEEPOLATION OF THE LOGAEITHMS (Xp^) BY THE METHOD OF FI-
NITE DIFFEEENOES 849—854
{a) Formulae when the intervals are (1) equal and (2) unequal 850 — 854
(h) The series \ls constructed by series of four orders of differences, of which the first is Xp.^?. 850 — 854
((?) Table of first differences 854
(d) Introduction of the factor v^ — 854
{e) In this form the series Xh can be calculated and printed by Scheutz's new machine . . . 854
lY. CONSTEirCTIOlSr OF THE COLUMlSrS 4, 4, L^, P^., Q^, T^, and notices of some of
their practical applications ; mean lifetime ; illustrations by Plate XLII 855 — 860
{a) Nature and uses of the new column Y^ ; mean ages of the population ; mean ages at death 857
(h) The determination of the value of sums of money dependent on the continuance of life ;
probabilities of life ; probable lifetime 858 — 860
MDCCCLIX. 6 S
838 BE. EAEE OH THE CONSTEUCTIOH OF LIFE-TABLES.
Page
V. THE THEEEEOLD LIFE-TABLE; (1) EEESONS, (2) MALES, (3) FEMALES ... 860
YL USEFUL FOEMULJE 860—861
VII. LIFE-TABLE OF THE HEALTHIEST ENGLISH DISTEIOTS 862—8
(a) General description of the districts 862 — 863
{h) Conclusion , 863—864
JSTote on tlie Two Hypotheses 867 — 868
TABLES :--
Table A. — Healthy districts ; population, 1851 ; deaths in the five years 1849 to 1853 ; rate
of mortality at different ages 864 — 865
Table B. — The values of the radical numbers and logarithms on which the two Life-Tables
for males and females were founded 866
Table B1 . — Logarithms of 4, stereoglyphed by the calculating machine 869
Table C. — Life-Table ; Persons, Males and Females, the living and dying at each age, or h
and 4 870—871
Table D.— Life-Table. Persons 872 — 873
Table E. — Life-Table. Males 874 — 875
Table F. — Life-Table. Females , 876 877
Note. — The three Tables, each containing the columns d,v, h, 1%, P.^?, Q*, Y^., constitute the
three-fold Life-Table.
Table G. — Tables of mean after lifetime (expectations of life) 878
Note. — The Tables were calculated in duplicate, and compared by Mr. F. J. Williams,
Assistant Senior Clerk, and Mr, J. Lewis, Junior Clerk in the Statistical Department
of the General Eegister Office. The logarithms of h were compared and found to agree
with those produced by the machine.
The Transactions of the Eoyal Society contain the first Life-Table. It was constructed
by Halley, who discovered its remarkable properties, and illustrated some of its appli-
cations. The Breslau observations did not supply Halley with the data to frame an
accurate Table, for reasons which will be immediately apparent ; but the conception is
full of ingenuity, and the form is one of the great inventions which adorn the annals of
the Royal Society.
Tables have since been made correctly representing the vitality of certain classes of
the population; and the form has been extended so as to facilitate the solution of
various questions.
In deducing the English Life-Tables from the National Returns, I have had occasion
to try various methods of construction ; and I now propose to describe briefly the nature
of the Life-Table, to lay down a simple method of construction, to describe an extension
of its form, and to illustrate this by a new Table representing the vitality of the healthiest
part of the population of England.
The Life-Table is an instrument of investigation ; it may be called a hiometer, for it
gives the exact measure of the duration of life under given circumstances. Such a Table
has to be constructed for each district and for each profession, to determine their degrees
of salubrity. To multiply these constructions, then, it is necessary to lay down rules,
DE. FAEE ON THE CONSTEUCTION OP LIFE-TABLES. 839
which, while they involve a mininium amount of arithmetical labour, will yield results
as correct as can be obtained in the present state of our observations.
1. aEKEEAL BESOEIPTIOF OE A LIEE-TAELE. (See Table C, p. 870.)
A Life-Table represents a generation of men passing through time ; and time under
this aspect, dating from birth, is called age. In the first column of a Life-Table age is
expressed in years^ commencing at (birth), and proceeding to 100 or 110 years, the
extreme limit of observed life-time.
If we could trace a given number of children, say 100,000, from the date of birth,
and write the numbers dovra that die in the first year, living therefore less than one
year, against in the Table, and on succeeding lines the numbers that die in the second,
third, and every subsequent year of age until the whole generation had passed away,
these numbers would form a Table of Mortality^ showing at what ages 100,000 lives
become extinct.
Again, if the 100,000 children w^ere followed, and the numbers living on the first, on
the second, and on every subsequent birthday until none was left, the column of numbers
would constitute a Table of Survivorship. So if of 100,000 children born at a given point
of time, the numbers dying (d^) in each subsequent year were written in one column,
and the numbers surviving (l^) at the end of each year in another column, the two
primary columns of the Life-Table would be formed.
It is evident that if one of these columns is known the other may be immediately
deduced from it; for if of 100,000 .children born 10,295 die in the first year of age,
3005 in the second year of age, it follows that the numbers living at the end of one
year must be 89,705, at the end of two years 86,700. Upon adding the column (d^)
from the bottom up to the number against any age (^), the sum will represent the whole
of the numbers dying after that age; and consequently the numbers living at that age^ as
shown in the collateral column (4).
The 100,000 children born at the same moment, and counted annually to determine
the numbers living at the end of every year^ would by our Table completely pass away in
less than 107 years. If another generation of 100,000, born a year afterwards, were
followed, the numbers dying in the various years of age would not be very different,
the circumstances remaining the same ; and the numbers of those entering each year of
age would vary inconsiderably from those of the first series. If 100,000 children again
were born at annual intervals, and were subject to an invariable law of mortality, they
would form a community of which the numbers living at each age would be represented
by the sucessive numbers (4,) in the Life-Table. The sum of these numbers, by the new
Table of Healthy Districts, would be 4,951,908. The births are here assumed to take
place simultaneously at annual intervals ; immediately before the births, therefore, in
such a community its population would be 4,851,908, to which it would fall progressively
from 4,951,908 by 100,000 successive deaths in the year. The average number con-
stantly living would be some number between 4,951,908 and 4,851,908; and it would
be very nearly the mean of these limiting numbers.
u S 2!
840 DE. PAEE ON THE COKSTEUCTIOlSr OF LIFE-TABLES.
In the ordinary course of nature, the births in a community take place in remittent
succession; and if it is assumed that the 100,000 births occur at equal intervals over
every year, it is evident that at any given date a certain number will be found living at
all the intermediate points of age between to 1 year, 1 to 2, 2 to 3, and all the remain-
ing years of age. The population in the above instance would be found by enumera-
tion to be nearly 4,899,665.
The annual births would be 100,000 in such a community. The annual deaths would
also be 100,000; and by taking out the deaths at each year of age, from the parish
registers of a single year, the second column [d^) of the Life-Table would be found. By
adding this column of deaths up and entering the sum of the numbers year by year
against every year of age (^), the third column {l^^) of the Life-Table would be obtained ;
for it has been already shown that the numbers attaining any age x are equal to the
numbers dying at that age, and all the subsequent ages. From the registers of the
deaths, a Table of the numbers of the population living in a parish so constituted could be
immediately determined without any enumeration. Its deviations from the truth would
be accidental ; and they would be set right by taking the mean of many years. So also
from a simultaneous enumeration of the numbers living in each year of age^ the two
columns d^ and l^ of the life could be constructed without reference to any registry of
the deaths at different ages.
The mean age at death in such a community would express the mean lifetime, or the
expectation of life at birth ; and the product of the number expressing the annual
births multiplied into the mean age at death would give the numbers of the popula-
tion.
The facts which a Life-Table expresses in numbers may be represented by the lines
of a figure ; age [x) being indicated by the abscissas measured from 0, the numbers
living (I) at each age by the ordinates of a curve line, and the numbers living between
any two ages by th^ plane surface within the two ordinates, the curve line, and the
corresponding portion of the abscissa. The relative numbers living at the ages 20 and
21 are seen in the two lines of Plate XLII. fig. 1, over the ages 20 and 21 ; if the
deaths in the intervening year all occurred immediately after the age 20 was attained,
the numbers living would also be represented by the parallelogram having its two sides
equal to the ordinate over 21, and for its base the portion of the abscissa between 20
and 21 ; but if all the deaths occurred only the instant before the age 21 was attained,
the height of the parallelogram would be represented by the ordinate over the age of 20.
The deaths occur at intervals between the two ages, so the numbers living, and the
lifetime which is passed between the two ages, are correctly represented by the curvi-
linear area.
The deaths in each year of age are called the decrements of life. They are repre-
sented by the differences in the lengths of the successive ordinates. Thus by cutting off
a small portion of the ordinate at the age 20, the ordinate at the age 21 is obtained;
this small portion, shown in Plate XLII., represents the decrement of life in that
year of age. It will be observed that the decrements vary at every year of age ; and
DE. FAEE cm THE CONSTEIJCTION OF LIFE-TABLES. 841
this is more evident when they are exhibited on the larger scale of Plate XLII.
fig. 2. The decrement in the first year is large ; in the first five years the decrements of
life are considerable; at the age of 10 to 15 they fall to their minimtim; slowly increase
to the age of 56 ; increase more rapidly until the maximum is attained at the age of 75 ;
then decline gradually to 85, and after that more rapidly until every life is extinct at
the age 107 by this Table.
II. PEINOIPLES OF CONSTEUOTION. THE FUNDAMENTAL COLUMN 4.
The conditions of the hypothesis upon which the preceding reasoning rests are never
precisely realized in nature ; in the first place the number of births fluctuates, increases,
or decreases from year to year, and the deaths fluctuate still more ; rarely equalling the
births in number. Immigration and emigration interfere. Under these circumstances.
Tables such as those which Halley, Price and others made from the observations
on the deaths alone are never accurate, and require correction to give approximate
results. If it be assumed that the law of mortality remains invariable, and that migra-
tion does not interfere, then the nature of the correction to be applied to a Table
framed from the deaths alone will become immediately apparent by an example. The
births increase in England. Let the annual births in a portion of the community be
doubled in sixty years, thus be 50,000 in 1796, and 100,000 in 1856 ; then the deaths
of persons of the age of 60 in 1856 must be doubled to obtain the deaths which
would have happened at that age if the annual births sixty years before these deaths
had been 100,000. If the births have been accurately registered, formulae for correct-
ing the ordinary Table drawn up from the deaths at different ages will be suggested by
the above considerations.
I now proceed to describe another method which has been adopted in framing the
Table C, and is applicable wherever (1) the number of annual births, (2) the numbers
of the population living at definite periods of age, (3) the deaths at the corresponding
ages during a certain number of years, in any community are ascertained by observation.
This method is not open to the previous objections.
The aim is to obtain equations which will describe the curve lines (Plate XLII.
fig. 1) of the Life-Table, in the most direct way ; and these equations may be deduced
from the determined rate of mortality at certain intervals of age.
The relative numbers living at two ages, 20 and 21, can evidently be found from an
equation which expresses the relation of the average numbers living and dying between
those ages during a given time. This can be determined very nearly ; for although the
ages of the living are not ascertained with exact precision at the census, still by taking
all the numbers living at the ages 15, 16, 17 years up to 24 and under 25, together,
the aggregate represents very nearly the numbers living in that decenniad of life. The
deaths at the same ages are obtained with at least equal accuracy from the registers of
deaths. By this process, and by extending the observations over five or more years, a
number of facts is obtained sufiiciently great to yield average results ; and it may be
842 BE. FAEE ON THE COHSTEUCTIOH OF LIFE-TABLES.
assumed that the ratio of the living at the ages 15 — 26 * to the dying in a year at the
same ages 15 — 26 represents the annual rate of mortality at the exact age 20. So also
the mortality rate at the ages 30, 40, 60 and other ages maybe determined. As observa-
tions grow more exact, and the facts are multiplied, the intervals of age may be dimi-
nished to 6 years, and ultimately to 1 year.
In determining the rate of mortality^ a given number of persons living a year is
considered equivalent to twice that number living half a year, or to half the number
living two years.
Thus if nd represent the deaths in n years out of a number amounting on an average
to P during the same years, then -pzizm^z the rate of mortality, or the proportions of
death in a year (always taken as the unit of time) out of one year of lifetime. It is
found from all the observations hitherto made on a large scale, that the rate of mortality
varies at every interval of age ; but at the same age it may for the present purpose be
considered invariable under similar circumstances.
m^ therefore varies in every moment of age ; but I have employed it to express the
mean annual rate of mortality during the year following the year of age ^, .-. ^'=m^,
where d^ indicates the deaths, P,, the year of lifetime, after the year of age x. The
m^. is the expression of the force of the causes that induce death, of the death-force, vis
mortalis; and its reciprocal — =% measures the forces that sustain life, the m^ vitalis.
vyix
The vital force under natural circumstances may by one hypothesis be sufficient to
sustain a whole generation alive for seventy or eighty years, and then suddenly collapse.
The Life-Table, if this hypothesis were true, would be represented by the parallelogram
in which the curve of the Life-Table is inscribed (Plate XLII. fig. 1).
By the hypothesis of DEMOiVREf the rate of mortality is such, that at the age of 20 one
in 66 living at the beginning dies before the end of the year, leaving 65, 64, 63, 62, 61
to enter on each year of age until at the age of 86 all are dead.
Upon this hypothesis the relative numbers living up to the age 86 form an arith-
metical progression : and the deaths in the equal times are equal out of the diminishing
numbers living. The rate of mortality increases on this hypothesis as age advances in
the same ratio as ^— ^: 1 ; where n is the difference between the actual age w and 86.
It is called the complement of life. The Life-Table, upon this hypothesis, has equal
decrements, and might be represented on Plate XLII. fig. 1, by drawing a diagonal line
through the parallelogram. Its deviation from the true curve on this scale is evident ;
but it is also evident that a series of straight lines, which would nearly represent the
true curve, may be drawn from point to point of all the ordinates.
If the causes of death act with equal intensity at all ages, they may be represented
by any simple external cause, destroying an equal proportion of the numbers living in
equal intervals of time. Thus, if 1600 men were distributed equally over ground where
^ By this 15 and under 25 years of age is tuiderstood/ and so in all similar cases,
t See Treatise of Annuities on Lives, Preface to 2nd Edition.
BE. FAEE ON THE CONSTEUCTIOlSr OE LIFE-TABLES. 843
they were exposed to certain dangers represented by successive discharges of musketry
which at every discharge shot down one-half of the numbers remaining, they would be
reduced successively from 1600 to 800, to 400, to 200, to 100, to 50, and so on ad mji-
nitum^ if a fraction of a living man could be conceived : the numbers living at each
year of age in a Life-Table would not decrease at these rates^ but they would decrease at
a constant rate if the dangers at every stage of life remained constant and equally great.
The numbers of the living at successive ages would be in geometrical progression, and
would be represented by the ordinates of the logarithmic curve.
The law of mortality can only be derived from observation, and it is found to be less
simple than either of these hypotheses implies. It can, however, be represented nearly
by equations at different periods of age. Upon inspecting Table A (p. 864), it will be
seen that at the age 55 — 65, which may be represented by the exact age 60, the mortality
is such, that 2162 women die in a year out of a number equal to 100,000 living a year ;
and the mortality, which is the ratio of the dying to the. living in a unit of time, here
set down as a year, is therefore m= '02162. Again, the mortality at the age of 70 is
•04992 ; at the age of 80 it is -11866, and at the age of 90 it is -26711. The mortaHty
increases rapidly, and is more than doubled every ten years. The four numbers differ
little from the terms of a geometrical progression, the logarithms of which have a con-
stant difference. Let the rate at which the mortality increases be r, and r^^=2-3116,
and the first term (m) be -02177; then a series of numbers will be formed differing
little from those which express the value of m at decennial intervals of age.
Values of m at the precise age x,-— Females,
Age {cc). 60. 70. 80. 90.
By observation . . . -02162 -04992 -11866 -26711
By hypothesis . . . -02177 -05033 -11633 -26891
Note, — It may be assumed that m at 60 is the mean value of m in its range from
^59^ to meo^ ; and so in other cases.
The annual rate of the increase of m from the age of 55 to 95 is r =1-0874; and if
m is the mortality at any age after 55, then m^=^mr''=z the mortality at z years after the
age at which m is taken. The common logarithm of r is =Xr= -03639.
The mortality (m) of males at corresponding ages is higher than the mortality of
females ; but the rate of increase as age advances is nearly the same.
The value of m for females at the age of 20 is -00765, and the mortality increases at
the rate of nearly one-seventh part every ten years. The exact value of r is 1-0149,
and Xr= -006423.
Values of m, — Females,
Age.
By observation . . .
By hypothesis
By these observations in the healthy districts the mortality [m) of men at the ages
15 to 45 is lower than the mortality of women at the same ages; yet during that period
20.
30.
40.
50.
00765
•00894
•00998
•01192
00760
•00882
•01022
•01185
844
DE. PAEE ON THE CONSTEFCTION OF LIFE-TABLES.
the rate of increase f is nearly the same for the two sexes. From the age of 40 to 50,
and 50 to 60, the mortahty of males increases at a rate intermediate between the rates
of manhood and mature age.
Females.
Limits of ages.
15 to 55 or 20 to 50 r=l-0149 Xr=-00642
55 to 95 or 60 to 90 r=l-0874 Xr=-03639
MaMi
ales.
15 to 45 or 20 to 40 r =:1-0148
55 to 95 or 60 to 90 f=l-0874
Xr=-00640
Xf='03640
The subjoined Table exhibits the series of values for m derived from the hypothesis
of two constant rates, and from direct observation. The values of r for females may be
evidently applied to males in every period, except in the ten years of age, 40 to 50.
Mortality (m) of males and females, (1) derived from observation, and (2) from the
hypothesis that m increases at the preceding rates.
Precise age.
Annual Mortality to 100 constantly living at each. age (m).
Males.
Females.
By observation.
By hypothesis.
By observation.
By hypothesis.
m
so
40
50
60
70
80
90
•691
•818
•928
l^273
2-294
5-486
12^817
28-350
•696
•807
•935
b083
2*329
5-385
12-451
28-785
•765
•894
•998
M92
^•162
4^992
11^866
26-711
•760
•882
1^022
M85
2-177
5^033
11 •633
26-891
100
40-000?
66-550?
45-000?
i
62-160?
The observations on the numbers living and dying of the age of 95 and upwards are
exceedingly uncertain; and it is probable that many of the persons believed to be 100,
&c., are really persons five or ten years younger ; so that these values of m„ by the hypo-
thetical method, are probably as correct as the direct numbers.
I shall now notice briefly the appMcation of this hypothesis, first suggested by
Mr. GoMPEETZ, and applied by him to the interpolation of the Northampton and other
Tables*. Mr. Edmonds, in 1832, extended the "Theory," and applied it to the con-
struction of three Life-Tables f. He gave an elegant formula, similar in principle to
that of Mr. Gompertz, from which the curve of a Life-Table can be deduced, upon the
above hypothesis.
^ PhilosopMcal Transactions, 1825, paper by B. Gompertz, Esq., F.E.S.
t Life-Tables founded upon tbe discovery of a Numerical Law regulating the existence of every Human
Being, &c. By T. E. Edmokds, B.A., 1832.
BH. FAEE ON THE CONSTEUCTION OF LIFE-TABLES. 845
5
In the equation -7=v, where s indicates space, t time, v velocity, the units of measure
must be fixed before numbers can be inserted in the general expression ; and then v will
express, in the measure that has been applied to space, the number of such units of
space described in one unit of time. Here v is a. ratio ; it is the rate at which the body
moves : and in the same manner m, in the equation j=m^ is the rate of dying^ that is,
as I shall express it, the mortality ; or it is the ratio of the dying to the living in a given
unit of time, the time during which the deaths occur being of precisely the same dura-
tion as the time during which the living are under observation,
I (living during 1 year) : d (dying during a year) : : 1 (year of life) : m.
If for I the number 100,000 is substituted, it is assumed that immediately a death
occurs another life is substituted; and as the time is a year, then 760 will represent the
value of ^ at the age 20, according to the preceding Table; .*. m= '00760. If the time^
instead oi one year ^ be the thousandth part of one year, then m=: -0000076; and if the
time be infinitely short, m will be infinitely small : m is a ratio ; the quantity of life
existing during the time is represented by 1, and the quantity of life destroyed by a
fraction, m. Whether the life inheres in the first organic molecule after conception, in
the infant, or in the man, the vital action has a certain force of continuance, which is
constantly varying ; and the amount of this force that is extinguished at a given instant
of time will be represented by the force of mortality, namely, by m at that instant.
Then let the age ^=;2+^5 where a represents the number of years up to the age at
which a given rate {r) of increase of m begins; then z=^x—a. And the mortality at
any instant of age, in an instant of time at the end of z years or parts of years, will be
mf". Now let y represent the living at that precise age ; then the decrement of y in an
infinitely short time will be —dy^=-ymr''dz\ the dy being negative as it is taken in a
direction opposite to that in which the ordinate y of the curve is assumed to be drawn.
Transferring y to the other side of the equation, this becomes =^mr''dz; and inte-
if
grating both sides, we have {\,y being put for the hyperbolic logarithm of y, and "K^c
for the difference between the constants of the two integrals) —
c mr^
Kc-\^=K-=— ; (1.)
^^,flf
>^J=V-T7' (2-)
and x^c=X^y+% . (3.)
At
m
"When z is made zero, let y=X; then X^y will also disappear, and K,c= — . Upon
substituting this value of X,c in equation (2.), it becomes
^^=3^-X7=v(l-^) (4.)
MDCCCLIX. 5 T
846
DE. PAEE ON THE CONSTEUCTION OF LIFE-TABLES.
Upon passing to the numbers, equation (4.) becomes
m,
(1-r^)
^--.gXgr —the value oiy (taken as 1 at the origin) at the end of z years.
Let \ denote the common logarithm with the base 10 ; then Xty=-#, where Tc is the
modulus of the common system of logarithms ; as also
km , mr^ kmr^
X.c=T-j and -—=:-- —
* Xr' Afr Ar
Equation (2.) becomes, after the required substitutions,
Ky km kmr^
and
k \r Kr
h^m
SO the equation becomes finally y=10>.^^^ *" ^
This is the form given by Mr. Edmoistds, and is convenient for use.
(6.)
(6.)
By making z successively 1, 2, 3, up to any number less than the number
of years of age within which r remains constant, the number 4 being known, the
number living at any other age within that range will be obtained by multiplying 4 by
the corresponding value of y. Thus, iiyi^ is the value of y when 2^=10 in equation (6.);
then putting 4o foi' the numbers hving at the age 20, the living at the age 30 will be
This hypothesis does not express the facts deduced from the observations exactly.
Ifm^ could be expressed exactly over more than 20 years by m^'=^m^T''^ the first difier-
eiices (h^) of the logarithms in the series following would in a certain number of cases
be equal.
Females in Healthy Disteicts of England,
Precise age.
Annual rate of
• mortality.
Logarithms of the
annual mortality.
First decennial
differences of
Second decennial
differences of
Xm^.
oc.
m*.
\m.
ai.
^K
go
30
40
50
60
70
80
90
100
-00765
•00894
-00998
•01192
3-8835
3-9512
3-9992
2-0762
•0677
-0480
•0770
-•0197
•0290
-1817
-1047
-2587
-02162
•04992
•11866
•26711
•45000
2-3349
2-6983
1-0743
T-4267
1-6532
-3634
•3760
•3524
-0126
•0236
--1259
•2265
* Here, at the age 20, w is the mean mortality that rules over the age 19^ to 20 J years of exact time.
DE. FAEE ON THE COKSTEUCTION OF LIFE-TABLES. 847
The inequalities in the second differences vary in every separate class of observations ;
but there is generally a tendency in the first and in the second diiFerences to increase,
over a certain extent of the series. The error of the hypothesis is slight if the rate of
increase (r), of which X -00677 is the logarithm in the case in hand, is only assumed to
remain uniform for the ten years 20 to 30, or for the one year 20 to 21. Now let the
number living at the age 20 be represented by 4o5 ^^d the number living at the age 21
I
by 4i ; then put -~=p.^Q. Here it is evident that if 4o ^^d p^^ be known, 4i is deter-
mined immediately by the equation 4i=4oXjp£o- But j?2o is the value of y in the equa-
tion ^1=10^''^^"' \ when z is put =1. Taking the numbers from Table A., we have
m=-00765 at the precise age 20=(19|+20^)i; and Xm= 3-8835130; Xr= -0067728;
and .-. r=l-015717 ; Jc is put for the modulus of the common logarithms,
.-. X^^= 1-2755686; Jc{Xr) is the complement of the logarithm of (Xr).
W 1-2755686
Km
3-8835130
k{Xr)
2-1692317
X(l-r)
2-1963697
-•0033472
3-5246830
1-9966528
As the factor (1 — r) is negative it makes the exponent of 10 negative, and upon
taking the complement of this the logarithm of y is found to be 1*9966528. This is
also the logarithm of ^20= *99232 ; and it enables us to pass, in the construction of a Life-
Table, from the living at the age of 20 to the living at 21. If we obtain the several
values jp^ at every year of age, the whole of the Life-Table can be constructed.
It will be found that p^ is always a fraction, and it does not differ very much from
1— m^. But while m/^ shows the deaths in a year out of a unit of life (which may con-
sist of any number of individual lives constantly kept up), p^ shows how much out of a
tmit of the same life at the beginning of a year, the dead not being replaced, survives
a year after the age w ; and 1 —p^ is the amount of loss which occurs in the same year
out of a imit of life at its commencement. Thus, as ^20= '99232, it follows that
l—j}2^=: -00768. In the same year of age 20 to 21 the mortality is m2o=*00771, or
-00003 more than (1— jpao)- If the unit of life is made 100,000 living at the age 20,
then 99232 will survive, and 768 will die in the ensuing year of age. But if it is
assumed that the deaths take place at equal intervals, it may also be assumed that the
number of lives (100,000) being constantly sustained, the accessions of 768 new lives
take place at equal intervals, consequently that they are under observation half a year
on an average, giving the equivalent of -y- =384 years of lifetime at the age 20 to 21 ;
* m serves to indicate the mean mortality in the year following the exact age ^.
6 t2
848
DB. FAEE OK THE COKSTEUCTIOK OF LirE-TABLES.
•99232*, as before.
now out of this number (384) at that age three die when the mortality is m^^. This
accounts for the difference of -00768 and -00771 ; the former occurring in a year out of
a unit of life of which the waste is not replaced.
From these considerations it may be inferred that if m^ is known, p^ may be deduced
from it upon the hypothesis of equal decrements through the year by the formula
^,==:;r~-f-^=^r7"~^'- Thus m^^ being -0077072, we have ttt^t^^^h^t^ •
r^ l-fl-m.^ 2 + m^ ^" ^ ' 1 "0038536
The X^2o by the previous method is 1-9966528, and by this method it is the same. By
either of the methods the value of p^ may be deduced for the subsequent ages, and
^205 JP305 i^4o ....•*• -JPso? JPioo wlU be obtaiued. These values are here given, and it will
be seen that the results by the two methods are nearly identical at all ages, except the
two last, when the observations themselves become less exact.
Females.
Age {x).
^--Kl-a^)-
20
30
40
60
60
70
80
90
T-9966528
•9960967
•9966263
•9946669
•9902049
•9773538
•9463182
•8809176
r-9966527
•9960967
•9956264
•9946676
•9902073
•9773557
•9462643
•8801776
It will be observed that the fraction ^==
1— l^m
l + iwi
approximates to 1— m as m becomes
less ; for upon developing it into a series, j9 = 1 ■— m + i^^ — h^^ + i^^^ • • • • ^^^ taking m
infinitely small, the terms after the two first may be neglected.
The values of mo, mi ...... mg may , be obtained by the method already described. But
it rarely happens that the population living at each year of age is accurately enumerated
at the Census ; and besides inaccuracies of statement, the numbers living at each of the
early years of age fluctuate considerably, so that the numbers of children living of each
year of age in 1851 do not represent the average numbers living of those ages in the
five years 1849 to 1863, for instance.
The following method is less exceptionable. It may be assumed for this purpose
(1) that the births registered in the year 1848 represent the births in that year ; (2) that
the births are equally distributed over the years in which they occur, and consequently
* ^ at tlie precise age 20 is nearly -00765. The increase in this mortality from the age 20 to 20|^,
the middle of the year of age 20 to 21 is obtained by adding |-Xr, as above given, to Xwig^, that is, to the
log of (wig4+^2o^)i; .% ^%o= '0077072
Xm,9i 3-8835130
iXr 00033864
\mm 3-8868994
BK. l^AEE ON THE COFSTEUOTIOIS' OF LIFE-TABLES. 849
(3) that the mean date of all the births in the two years 1848, 1849 was immediately
before January 1, 1849. The half of the births in those two years will consequently
represent pretty accurately the number of births out of which the deaths of children
under one year of age happened in the year 1849. And the deaths and survivors cian be
followed by this method year by year, as is evident in the annexed scheme : —
Age
rj (births 1848, 1849)=mean annual births of which the mean date is January 1,
j [1849.
iminus deaths under age 1 in 1849
1 = surviving on' January 1, 1850.
minus deaths age (1 to 2) in 1850
2 = surviving on January 1, 1851.
minus deaths age (2 to 3) in 1851
3 = surviving on January 1, 1852.
minus deaths age (3 to 4) in 1852
4 = surviving on January 1, 1853.
minus deaths age (4 to 5) in 1853
5 = surviving on January 1, 1854.
By commencing with the mean number of births in the years 1849, 1850, and deducting
the deaths, a similar series may be obtained ; and thus a succession of similar series
may be deduced, the mean of which will supply the ordinary series 4? k^ 4? 4? 4? 4 of
a Life-Table.
These series are liable to various disturbances. If all the births are not registered,
the rate of mortality is overstated. If all the deaths are not registered, or if the chil-
dren are carried off as emigrants, the decrements of life are understated. The annual
number of births fluctuates, and now increases in England; they are in excess also in the
early months of the year. Several of the disturbances are slight, and some of them are
in opposite directions. The results can also be, and have been, checked by the results
of the other method. The value of m>j and m^^ are deduced by dividing the annual
deaths at the ages 5 to 10 and 10 to 15 by the mean population at thos6 ages. The inter-
polation of the series Xp^ from Xj?3 to Kp^^ succeeds; taking Xp^, Xp^, Xpig, and Xjjgo ^^ the
fixed points of the series, and Xpig being adjusted to allow for the turn of the curve.
The Tables A, B, and C supply the data from which the Life-Table of Healthy
English Districts was deduced. One or two arithmetical examples of the application of
the method adopted in the earlier ages are also supplied.
III. INTEEPOLATION.
We have therefore determined the values of >^^ at certain ages. The values of X^^, at
the intervening ages may be determined by changing the value of r, and making z suc-
cessively 1, 2 10 in the formula (p. 846). They may also be interpolated for every
year of age by the method of finite differences ; and upon the whole this method is
850 DE. FAEE OK THE CONSTEITOTIOK OF LIFE-TABLES.
preferable to any other. The logarithms of j?.^ are required; and to them it will be con-
venient to apply the interpolation directly. Any number of differences beyond four
becomes cumbersome, and it will be therefore sufficient to give the general formula,
which can be employed in deriving the first of either four or three orders of differences.
Investigation of Formulw — -Intervals equal.
Let any numbers of a series be so related that t^^, the ^th from the first, t^o, is deter-
mined by the equation (1.)-
—^^^jO 4. 1.2 ^ - 1.2.3 ^ + 1.2.3.4 0. . . K^O
l\ B^, S^ and S^^', the first differences of the four orders, are unknown; they can all be
determined from any five values of %. Now let n be successively 1^, 2^, 3^, 4^ ; then
the coefficients of ^o? ^1^5 '^hx'^ '^3^5 ^4^ can be found, to give the values of ^\ ^^ S^ and V- in
four equations. But when w is ten or more the coefficients become large, and the nume-
rical calculation laborious. It is therefore well to obtain the numerical values of
S*, S^, S^, V- in succession. Thus if the series is ascending or descending, the following
are convenient forms. The upper rows of signs are used in the ascending^ the lower
rows in the descending series : —
'^n-
90^
(2.)
■ ■ 5 ~~ t^ "o'l w ' " JL iC/ . . . . • • . t . . • 10. I
l^^ + ''' - r + '° - {,-1)1^ I i^^z:l^±li) l^ (4.)
w" + 12
8 = + ^- S _ -6— S + 24 ^ •
(5.)
It is necessary to be careful in deducing the successive values of I from the values pre-
ceding ; and before commencing their use their accuracy should be tested by inserting
them in the checking equation,
— +lf >5i + 4^(4^"~1) V2 + 4^(4^-- 1) (4^—2) V3 -|_ 4^(4^— 1)(4^-— 2)(4^— 3) ^^ .^ .
X may be any number. If only four terms are given, §^ is assumed to be constant ; and l^
being 0, all the terms into which it enters disappear. The above formulae, if this is
borne in mind, are applicable when S*, S^, or S^ are assumed to be constant, and serve
therefore to supply the differences when there are one, two, three, or four orders by the
most expeditious method.
* It will be borne in mind tbat these imply first differences, or ^^j^^, ^%q, ^%^, ^^Uq,
DE. FJlKR on the CONSTETJCTION OF LIPE-TABLES.
861
In constructing the Life-Table, a; was made 10 from the age of 20, and on inserting
the numbers, the equations (2, 3, 4, 5, 6) became
+
U40
'4^30 j" 6^20
4uio + %
+
%o
10,000
Xo'wO •
(7)
s=
iM20
1000
2^10 "J" %
+
• •••••»•••• I o» I
100
9S
3 —
+ -
A.A3Jk^
• • • • • it/«i
l'=- + 7 44S' ~ 12§'T 21S\
10 + 2 - +
• • •
. (10.)
The checking equation is
u..— : u. + 40^^ t 780^^ + 9880^^ t 91390^^ .
_ +
^40 -|- "^0
+
+
• '•
• •
• I J. J. « /
If three orders of differences are used, the checking equation is
t t^o ^ 30^^ t 435S^ + 4060^^
+
(12.)
After adding or subtracting any constant to or from a series of numbers, the differences
remain the same ; and if consecutive terms are multiplied or divided by the same factor,
the differences are multiplied or divided by that factor. Thus (b-^a)'—(c-\-a)=^b—c^
and ab^ac=a(b^c). Advantage is taken of these properties to reduce any one of the
terms in the equations to zero.
Thus let the logarithms to be interpolated be the following — values of j?2o? jPso? ^40? ^nd
jp^Q, taken from the column headed males, Table B ; then they may, among other ways,
be interpolated as follows : —
As 1-9969724 is the contracted expression of (•9969724— 1), we have
Age __
20 1-9969724:
30 T-9964260:
40 1-9959051:
50 r-9943048
•0030276
•0035740
•0040949
•0056952
(1) Multiplying each term by 10,000,000,
that is, striking out the decimal point
and the two adjoining ciphers, and (2)
then subtracting from each 30,276, the
values of u^^zXp^ to be operated on
. become
Mo =
-00000
w,„=
- 6464
%o=
-10673
w^<.=
-26676
By inserting these values with their negative signs in the equations, and taking the
upper signs, the three differences are found ; that is,
S3__ii.049: S2_ioi-991; and §^=-872-7715.
The differences are now divided by 10,000,000, that is, ciphers are added to their left-
hand side, so that the above decimal point may be moved seven places in that direction.
852 DE. PAEE ON THE CONSTEUCTION OF LIPE-TABLES.
and the operation may be thus commenced. By adding the differences successively to
each other and to X^2o= 1*9969724, the successive values are found of 'kp2\^ Xpa?? ^^23 •
Xpso ^P to and including Xpsg for males, where the series joins naturally the subsequent
series, commencing at \p^^,
^\ P. ^\ _ Xp,,
--•000,0011,0490 -000,0101,9910 --000,0872,7715 1-996,9724,0000
(constant) -000,0090,9420 -~ -000,0770,7805 T-996,8951,2285
- -000,0679,8385 1-996,8180,4480
T-996,7500,6095
In the actual operation the S^ is subtracted from S^ S^ from S^, and §^ from X^^ ; it is
therefore convenient to substitute for their present values the complements of S^ and S\
as thus all the series become additive.
As X?2o+>^i>2o=>^4i5 and X4i+>^_P2i=^42!) ^^d generally XZ^.+Xp^.=X4+i, it is evident that
the Vp^ is the ^rst difference of the series \l^\ and the whole series, XZ^., from X^o to X?5g,
may be formed as in the subjoined example, where S^ becomes S'*, V" becomes 5^, and
so on.
Healthy Districts. — Males.
S* (constant)
9-999,9988,9510
9-999,9127,2285 9-996,9724,0000 4-584,1951,2769
9-999,9229,2195 9-996,8851,2285 4-581,1675,2769
9-999,9320,1615 9-996,8080,4480 4-578,0526,5054
9-996,7400,6095 4-574,8606,9534
4-571,6007,5629
JSlote, — The four last figures in the decimal portion of the series >jp^ and in Td^ may in
practice be omitted.
The corresponding values of Xp^ in the column headed Females, Table B, are inter-
polated in the same way. And the X^go? ^^i^7o? ^i^so^ and Xp^o are interpolated by the
same methods, the series being continued backwards to Xj^g^ and forwards to Xpios ; the
actual observations of age after the age of 90 furnishing results less reliable than those
thus obtained, which bring a generation of 100,000 to their last end in 107 years.
The successive values of X^^, in the period from the age of 3 to the age of 19 inclusive,
are derived from Xpa, Xp^, Xpi25 and Xpao, which represent u^, u^, Ug, and u^^. As the
terms of the series are here at unequal distances, the first differences cannot be derived
from the preceding formulae. The $ can in this and similar cases be derived from the
proper equations by substituting figures for letters. But three literal equations supply
formulae for finding the three first differences from any four terms of series of the kind
which have been discussed: Uq^ which has a troublesome coefficient, can always be
Age.
20
0-000,0101,9910
21
0-000,0090,9420
22
0-000,0079,8930
23
24
DE. FAEE ON THE CONSTEUCTION OP LIFE-TABLES.
853
reduced to zero, and is therefore omitted. The first given term being %, let the second
M, be the a^th from %, and %hy be the ^th, u^ the 2;th from u^. Here x<y<z. Then the
following equations give the differences*: —
V
6{(2/-^)5-(^-^)|^ + (^-y)5}
{y-x){{z-\){z-2)-{j,-\){y-2)}-{z-y){{y-\){y-2)-{x-l){x-2)}
S^=,-|:^{5-5-{(^-l)(j/-2)-(x-l)(..-2)}|} . . . .
U,
5^2 S?3
f XOi I
I ^ jC,* I
(15.)
By making 2^=2^, and 0=3.3;, these equations assume the same forms as equations (3.),
(4.), (5.), with the term l^ struck out.
Putting ^=4, 3^ = 9, and ^=175 the three preceding equations become those which
were actually used in constructing the series p^ to p^^ : %^ is reduced to zero and is not
used.
V3 45wj^— 221Wg + 306% ClfW
13260 ^ ^
^ ' 90 '
t^jj,='M()+17S^-j-136S^+680Sl
/-I H- \
• » • » • • • • • • • • « •• • IXi./
• • • • • « ♦ • » . . * • . • * CXO.i
Checking equation.
* • . . . • . * • . • . • « IJLt/.f
^ A useful Table in applying the above formnlse.
^.
(^_l)(^-2).
#.
(^-l)(^~2).
#.
(#-l)(^-2).
w.
(#-1)(a-'~2).
20
34S5
30
812
40
1482
50
2362
21
380
31
870
41
1560
51
2450
22
420
B2
930
42
1640
52
2550
2B
462
33
992
43
1722
53
2662
24
506
34
1056
44
1806
54
2756
25
55f
35
1122
45
1892
55
2862
26
600
36
1190
46
1980
56
2970
27
650
37
1260
47
2070
57
3080
2S
702
38
1332
48
2162
58
3192
29
756
39
1406
49
2266
69
3306
MDCCCLIX.
5 u
854
DE. FAEE ON THE CONSTEUCTION OF LIFE-TABLES.
Table of first differences in the Life-Table of Healthy Districts of England.
Males.
Age
3
m
59
60
U
'X-
4'631,5849,0000
4-084,1951,^769
4-403,7768,0454
4-394,3905,1434
V^=^'-
9-993,2422,0000
9-996,9724,0000
9-990,6137,0980
9*989,5894,0000
^2.
0-001,2416,1260,934
9-999,9127,2285
9-998,9756,9020
9-998,9460,9820
dK
9-999,8012,4393,666
0-000,0101,9910
9-999,9704,0800
9-999,9547,5320
^\
0-000,0141,9648,567
9-999,9988,9510
9*999,9843,4520
9-999,9843,4520
Note.— The last series p^^ was carried backwards; from Xj?^q to Xpg
"A
>(20.)
Females.
3
20
57
60
4-623,2586,0000
4-570,6868,3846
4-405,2189,6826
4-381,2818,8126
9-993,2928,0000
9-996,6528,0000
9-992,9332,3725
9-990,2049,0000
0-001,2164,1598,794
9-999,9241,5455
9-999,0836,2675
9-999,0720,4825
9-999,7874,2556,561
0-000,0060,2930
0-000,0123,2100
9-999,9637,7950 .
Note — -The last series px was carried backwards from Xp^^ to Xpr^
0-000,0170,4566,365
9-999,9994,2530
9-999,9838,1950
9-999,9838,1950
A series of the form '?^%+^'^^^4+i+'^''"^^4+2 is required in rendering the Life-Table
applicable to the solution of questions in Annuities and Life Insurance.
The logarithms of the series are obtained by making the first term of the new series,
}v{v%)^ and the first term of the first order of differences 'k(vj)^^)=zXV'-{''kp^=l\ the S^ l^
and l^ of the original series remaining unchanged. Taking the interest of money at
1 —
3 per cent. v=YrQg; and X'?;=:l-9871627,753.
The derivation of the new series from this value of Kv^ and from the above Table
(males), is shown in the annexed example. Any value of v"" may be introduced in the
same way.
§^=9-9999988,951
4-3274506
-3115858
•2956337
•2796045
•2635074
In describing the first English Life-Table, I ventured to express the belief that the
chances of life may ultimately be calculated by Mr. Babbaoe's machine^. Mr. Bab-
bage's conception has been realized in the original and ingeniously constructed machine
of the Messrs. Scheutz, which was favourably reported upon by a committee of the Eoyal
Society. The first differences to be inserted in the machine can be immediately deduced
from those given above; and we may hope ere long to see the logarithms of Life-
Tables, for single and for joint lives, printed from types cast in moulds stamped by the
machine now in the course of construction by the Messrs. Dofkin, for Her Majesty's
Government, at the instance of the Eegistrar-General.
^ Letter to tbe Eegistrar-General, in Appendix (p. 352) to his Fiftli Annual Eeport, year 1843.
S*=9-9999988,951
Age.
S'.
s\
X(«2?.j,) = fl\
20
0-0000101,991
9-9999127,2285
9-9841351,7530
•0000090,942
•9999229,2195
-9840478,9815
•9999320,1615
•9839708,2010
•9839028,3625
/
DE. FAEE ON THE C0N8TEUCTI0N OP LIFE-TABLES. 855
IV. CONSTEUCTION OP THE COLUMNS rf^, 4, L^, P^, Q^, T^, AND NOTICES OP SOME
OP THEIE PEACTICAL APPLICATIONS.
The series 4 has been constructed ; and from that series others are deduced to com-
plete the Life-Table, consisting now of six columns,
(1.) c?^=4--4+i=^^i^ber of deaths in the year of age following, out of l^ alive at the
age w. By taking w successively at 0, 1, 2, 3, to the last age in the Table, the numbers
di/inff in every year of age are obtained. The numbers dying of the age w and under the
age 4+w ^1^6 immediately derived from the column 4; as (2.) 4— 4+n=^*+^*+i • •• d^+n-v
When ^+^>a;=the oldest age in the Table, 4=^^+^a+i •••+^(o-
(3.) L^=4+4+i +^w' The series is formed by the successive addition of the
series 4? from ^ upwards.
^nci [0) X;^= — 2 —
P —7 —1/7 —7 4-J^^
The series in column P_,, is constructed from the two columns l^ and d^^ or from the
single column 4? as 2P^=4+4+i; ^^^ •*• P^=:- o'^"^\ •*• 4'==2P^,— ^^.^ ; so, conversely,
the series 4 can be constructed from the series P^. The P^ is assumed to represent the
population, as expressed by the Life-Table, living at the age w and under the age ^+1-
Thus P2o= the population of the age 20 and under 21 years.
By substituting the successive values of P^ in the equation (5^), P^+P;,+i ... P^+w, we
nave ^'a? i 'a'+i •• • • "t^-jj+w r ■2^^+w+i*
V^V ^^' -^^ 1-^^ + 1 "F-*^* +2 •••• -*-^+w—l'l-^
^x+n -*- x+n'i -^ x+n + li" -^ a?+n+2 • • • • i* -^w*
( / . I . *. v=c^— "• v=CA'4-}i-— — ^^^\n — ■*- * n~ -*- 0? + 1 "i -*- x+2 . . • • ^x+n—i* J-Uc coiumu Vc^^ IS constructeci
by adding up the column P^, and transferring the successive sums to the column Q^,.
By substituting for the series P^ its values in 4, we have
And by again substituting for the series 4 its corresponding values in c7^, we have
(9.) Q^= 2^^"hl2^^+iH~'^¥^^+2 — ~l"(^^"f"¥)*^to-
(10.) Thus Q^, is equal to the numbers dying in each year of age after the age ^,
multiplied by the time (expressed in years and fractions of a year) that they have
respectively lived over that age; and if ^=0, then Qo=^67o+l2^i+2^(?2 (^+i)^^+«?
when (^+^) becomes >6;.
(11.) This column Q^ represents, therefore, two distinct orders of facts: it represents
the sum of the number of years that will be lived after the age x by the 4 persons then
living, and .-. -7^= the mean after-lifetime; of which -f^ will be enjoyed before the age
x-^n is attained, and -7^ after the age^+7^is attained. At birth the mean after-life-
time is -p, the unit here being one year of individual life.
5u2
856 DB. FABE O'N THE CONSTBUOTION OF LIFE-TABLES.
(12.) Q^ also represents the sum of the numbers of men or women living at all ages
over the age x^ out of Qo living at all ages, as Q^ is in all cases the sum of the numbers
living in each year of age, represented by the series P^,. The unit is here an individual
Ddan.
(13.) Thus, on referring to Plate XLII. fig. 1, the lifetime of 100,000 children
born simultaneously may be represented by 100,000 parallel lines, drawn from AB
horizontally in the direction of CD until they cut the curved line BC. And Q^ is
the sum of these lines expressed in the linear units of the scale on the line AC ; so
T^^IQOQQQ^ 100 000 =48'99665 ; the mean length of those lines = the number of
years of mean lifetime.
It will be observed that in this Table, instead of 100,000 lines, these lines are thrown
into 106 groups, each comprising the variable number of lines terminating in each of
106 intervals numbered on the line AC, and representing years of age. And in these
short intervals it is assumed that the mean length of the lines terminating in the
eleventh interval (10 to 11) is represented by 10|^, and so on.
The relative numbers of persons living simultaneously at each interval of age will
also be represented in the same Plate, fig. 1, by 106 successive vertical lines, raised
from nearly the centre of each interval between the ordinates on the line AC, and
measured in units of which the line AB contains 100,000. The same lines bound the
figure representing the two orders of facts ; and the numerical units expressing the
aggregate length of the vertical lines equal in amount the units expressing the aggre-
gate length of the horizontal lines expressed in the horizontal units.
(14.) I will now explain briefly the nature of the column Y„ which I have added to
the Life-Table*. The Life-Table (column P^) exhibits a representative population,
such as would be constituted by separating every year 100,000 births as they occurred,
* See paper in Appendix to Begistrar-Greneral's Sixth Annual Beport, pp. 544-552.
JEcotract from the Begistrar-GeneraVs Swth Annual Beport (1845), p. 528.
''Mte.—RA-LJMY'^ Table (1693) contained tlie column P. Johe^ Smabt made 1000 "bom" the basis
of his Table (1738), and introduced the columns d and Z. Simpsok adopted Smart's form of Table, which
was followed by Keesseboom (1738), Bepaeoietjx (1746), Price (1773), and Miliste (1815). The
columns S.y, j/ and Ay in Dutillarjd's *Loi de Mortalite' (en France) dans I'etat naturelf,' correspond
wdth the columns L, I, d in the new Table. The S.y added by Dutillabd is our L and Baeeett's column
B ; Dutillabd' s short Table (p. 123) has the four columns d, I, P, Q for quinquennial or decennial ages,
and the 'expectation of life.' Mathieu's Table II, is an expansion of the column Q of Dutillabd' s
short Table, and is that column for each year of age. In a recent report on the Bengal Military Pund,
Mr. Daties has a Table (1) containing columns corresponding with the d, I, L, P, Q of the English Table,
the ' Mortality per cent.,' and the ' Expectation of Life' at each age J."
I haTC in this paper employed d, Z, L, instead of C, D, N, which have been formerly used by me and
others, and should still be used where the factor -y^ is introduced.
t Influence de la Petite Yerole, p. 161. J See the note (A), p. 568.
DE. FARE ON THE CONSTEUCTION OF LIFE-TABLIS. 857
and keeping them apart in a separate community, subject to a definite law of mortality.
Any population living in the tabular proportions at each year of age may, for the sake
of distinction, be called a normally constituted population.
The ages of the population represented by the Life-Table amount, in the aggregate,
to Yo years ; it is the aggregate number of years which they have already livedo and, sin-
gularly enough, it is also, if the law of mortality remain constant, the numher of years
which they will live. Thus Qq persons in such a population have lived on an average
Y
^ years ; that is their meaf age, and it is also their mean after4ifetime. Y^ is the
number of years that Q^ persons have lived over the age w; and the mean age of such
Y Y
persons is ^+^^; their after-lifetime is g^.
The series Y^, is formed by successively adding up a series of the form ^^(Q^ H-Q^+i),
commencing at ^'+l=^=the oldest age in the Table.
(15.) .*. Xo==^Qo'T"yi I ^2 •• • "1^^(05
By substituting for Qq, for Qj, for Qg, and so on, their values in P,,, it will be found
that
(16.) Yo=:iPo + liP. + 2iP,+ 3-IP3 .... +(^ + Wn .... +(^ + i)l\^
(17.) But the mean age of the persons (Pq) of the age of and under 1 is nearly ^ ;
and so the series |, 1 J, 2|, 3|^, 4^, 5^, 6| (^+i) expresses nearly the mean age of
all the persons in the first (Po), second (PJ, third (P2), and (^^+l)th (P,J years of age,
and so for all other ages; consequently the sum of the series (16) Y,, is the sum of the
ages of all the persons living contemporaneously, as they are represented in the Life-
Table.
In like manner it is shown that
(18.) Y,=iP.+ (l+i)P,,,,+(2+i-)P,^,.... +(^+.i-«^)P^
is the sum of the number of years that the Q^, persons in the Table have lived over the
Y
age X. They have all lived x years ; and consequently ^+nr S^^^^ their average age
Y
precisely as g^ gives the average age of the whole community.
(19.) It has been shown that Q,, expresses the number of years that 4 persons will
live ; in the same manner it may be shown that Q^^. ^ expresses the number of years that
4+1 persons will live; .\ (4+4+i) persons will live (Q;^.+Q,+i) years, .-. |(4+4+i)=P^
persons will live ^(Q^+Q.,?+i) years. xAnd the same may be demonstrated for each
successive value of x.
But the sum of the series P^ is Q^.=: the number of persons living of all ages. And
the sum of the series i(Q^ + Q^+i) is Y^= the number of years that Q^. persons will live ;
.-. ^'=the mean after-lifetime of all the persons living simultaneously of the age x and
upwards. Thus by the Table D, 4,899,665 persons are living contemporaneously; their
mean age is q^= 4099555" = 33-92 years; and they will live on an average 33*92 years.
858 BE. TAEE ON THE CONSTEUCTIOK OF LIFE-TABLES.
(20.) The Life-Table serves to determine the value of Life Annuities, the value of
policies, and the premiums of insurance.
This is effected by introducing a new unit, such as £1, 1 franc, 1 dollar, or any other
monetary unit. Thus if £1 is payable at each death, the series d^ will show the number
of pounds falling due in each year of age ; so if £1 is payable by each person on attain-
ing the age ^, and each subsequent year of age, the series 4 shows the number of pounds
payable every year by the 4 persons ; and N^ will be the number of pounds payable in
N £1
the whole course of life after the age w : thus -4 — = the average amouj^t of an annuity
of £1 payable on each life at and after the age w. The money-unit may be introduced
into the other columns ; and ^ • £1 would show the average amount payable under an
annuity of £1 on each of Q^ lives. The present value of these future payments can
always be determined by assuming a given rate of interest. The estimates thus obtained
are also always read subject to the qualification that by hypothesis the Life-Table is
based on a law of mortality actually to rule for a definite time in the population to
which it is applied. The probability of the hypothesis is not here in question.
Under the same circumstances masses of mankind appear to experience, at the same
ages, the same rates of mortality. Consequently if for several years d^ persons have
died annually on an average out of 4 persons living at the beginning of the year, other
things being equal, the probability that the same number will die out of 4 persons in a
year to come is greater than any other that can be named, and the fraction expressing
that probability is j-- We know that d^, expressing the numbers dying in a year, 4+i
must express the numbers surviving as 4+i + 4=4- The chances may be represented
by 4 balls; 4+i white balls in an urn will represent the chances of living, 4^ Mack balls
in the same urn will represent the chances of dying. Now let each of 4 persons pay the
sum z for a ticket, and each person that draws a white ball be entitled to £1. Before
the drawing commences the value of each ticket is -y-*; for 4 (the total chances): 4^.^
(the chances in favour of winning on one ticket) : : 1 : '^=z.
Put 4=30,007, and 4+1=29,647; then ^^:^=z:^^~^p =£-98802. The amount of
money to be paid on 4+j white balls is £29,647, and £-9802 x 30,007=^. 4=:£29,647.
In like manner it may be shown that if £1 is paid to each person who draws a black
d £\
ball, the value of each ticket is -|— =:j/£l ; for ^. 4.£1 = 4£1, and £1 is to be paid on
each of d^, tickets.
Should £1 be paid alike to those who draw white balls and to those who draw black
balls, the value of a ticket will be equal to the sum of the two fractions expressing the
several probabilities, namely.
DE. PAER ON THE CONSTEUCTION OE LIFE-TABLES. 859
As one or other of the two kinds of balls must by hypothesis he drawn^ and £1 is paid
for each ball, the receipt of the £1 is certain : certainty is thus in all cases expressed by
unity.
If every ball as it was drawn were replaced in the urn, although in 30,007 trials
29 647
white balls were not actually drawn 29,647 times, black balls 360 times, still ^qqqh
would express the probability of drawing a white ball, and the value of £1 contingent
on that event, more accurately than any other fraction that could be named.
Again, if an urn contained by hypothesis an indefinite number of balls, out of which
29,647 white balls and 360 black balls were drawn and then replaced, the probability
of again drawing a white ball on trial, and the value of £1 contingent on that
29 648
event, would be expressed more accurately by oqqqq ^ than by any other fraction that
could be named ; past experience being by hypothesis the only means we have here of
judging of the future.
Thus a Life-Table applicable to the case furnishes the fractions to determine the
value of any sums of money dependent on the life or death of a given person, or a
certain number of given persons in a given time.
The probability of living two years expressed by the fraction -P=~ — ^, — , is
less than the probability of living one year.
Making n any number of years and fractional parts of years, the fraction -^ will
invariably express the probability of living n years after the age w. As n approaches
zero the fraction will approximate to 1, the symbol of certainty ; thus a person is more
likely to live a day than a year, a minute than a day. As n increases Z^+^ diminishes
in value; and when w-\-n expresses a year after the age a; in the Life-Table, l^^^ is by
hypothesis zero, .*. ^=:t-=0. The chance of living so long is expressed in this case
^x "x
by zero, the chance of dying in the time by 1, the symbol of certainty.
(21.) ?^+,, expresses the number of chances in favour of surviving n years, and l^'—l^+n
the number of chances of dying in the same time, the sum of the two together (l^)
expressing the total number of chances. Thus the fraction / ^ \ expressing the pro-
bability of living a given time ranges from 1 to 0, and ^— j^^=l— --y^, or the chance of
dying in a given time also ranges from 1 to as ^ varies. When the two fractions are
ecjuai —J = —J , tlien ta,^n^^^^x '«+w5 ^nci ^^^,+^=^^-5 •*• ^it?+»~^2*
To verify the equations, an age x-^n must be chosen at which ^^+„ is exactly equal to \l^.
Thus by the Life-Table of healthy districts 100,000 children born alive are reduced to
50,851 in 58 years, and to 49,895 in 59 years; so the chances are rather in favour of
^ The addition of 1 to the numerator, and of 2 to the denominator, may be neglected, when, as in this
case, the numbers are large.
860 DE. EAEE ON THE CONSTEFCTION OF LIFE-TABLES.
their living 58 years, as they are 50,851 to 49,149; upon the other hand, the chances
of their living 59 years (49,895) are less than the chances 50,105 of their dying before
attaining that age. Upon trial it vrill be found that the chances of living to and the chances
of dying before 58|f^ years=58+ — - — ^ — ' — ^^^+956 1^^^^^ ^^ about 58f years
are nearly equal ; hence this is called the prohable lifetime^ or vie ]prohahle by French
writers, for -^=:-. At the age 20 the probable lifetime is 47-x-|ff? nearly 48 years.
The probable lifetime at every age is immediately seen by inspection.
(22.) Y. THE THEEEFOLD LIFE-TABLE— PEESOJ^S, MA.LES, FEMALES.
The Life-Table is threefold. A Table having the six columns is made for males ;
another Table is separately made for females. The several columns of the two Tables
incorporated together form the Table of persons which has 100,000, and may have any
other number for its basis. The basis of the Male Table in the illustration is 51,125,
while the basis of the Female Table is 48,875. In that proportion males and females
were born in the districts. Under this arrangement the number of contemporaneous
males and females living at each age in columns 4 is shown: thus 38,388 males and
37,212 females attain the age of 20; 17,145 males attain the age of 70, and 17,133
females attain the same age ; at all ages under 71 the number of males exceeds the
females; at the age of 71 and upwards the females exceed the males in number: and
upon referring to the columns ^^., it will be seen that the males die off in greater
numbers than females after the age of 42. The age after the second year at which the
greatest number of deaths occurs is 75 in males, 76 in females.
These numbers all refer to the Life-Table for healthy districts.
Some of the other properties of the Life-Tables, admitting of innumerable applica-
tions in the solution of social phenomena, will appear in the following formulae, which
will be found useful in practice.
Yl. USEFUL FOEMIJLiE.
The following formulae will facilitate the use of the Life-Table. The figures must be
taken from the Tables of Persons, of males or females, applicable to the case. The for-
mulae are general, and are applicable to any other Life-Table.
d
(23.) ™=w^^=the rate of mortality in the year of age following the precise age w.
X
(24.) y==^~j^^^=l— -y^^rzithe probability that a person A of the age ^, in average
health, will die in the following year.
(25.) •Y^==^^=^y-^==:l--~==the probability that A, a person of the age x^ will live
a year; .-. 1— -^^^the probability that A, age ^, will die in the year following^ as cer*
tainty of life =1.
DE. PAEE ON THE CONSTEUCTIOISr OF LIFE-TABLES. 861
(26.) - ,'^'^" =the probability that A, age oc, will die in the next n years.
(27.) ^= the probability that A, of age x^ will live n years.
I
(28.) Put~=4+w; Q'^d when 4+^ is taken at such an age as to fulfil the conditions of
the equation, then n is the jprobable Ufetime=vie probaMe=t]ie time that it is an even
chance a person of the age w will live.
(29.) -7^= A^=:the mean after lifetime^ or as it is often called, the expectation of life —
an incorrect expression, which is rather applicable to the probable lifetime.
Note. — Upon Demoivre's hypothesis, the probable lifetime^ that is the time that a
person may fairly expect to live, his expectation, was the same as the mean after lifetime.
(30.) G^=:^+A.^=the mean age at death of persons who have already lived exactly
X years.
(31.) S=c — j^=the number of members of any Society between the ages x and x+n,
which will be permanently sustained by c . . . annual admissions at the age x,
S/
(32.) c=Q-^=annual recruits of the Society (S).
(33.) Q^^=annual members leaving the Society (S) on attaining the age ^+^.
(34.) Q^^== annual deaths in such a Society (S).
(35.) SQ^^=the aggregate number of persons living, who have left such a Society,
as pensioners or otherwise.
In the following formulae it is assumed that the population is normally constituted.
Y
(36.) q''= A^=the mean after lifetime of all persons of the age x and upwards.
(^'^O Q^::rQ^"''==Q^^=*l^^ mean after lifetime of all persons of the age of x and
under the age of ^+^.
Y
(38.) c.7^^^=the number of persons of which a Society will ultimately consist.,
recruited by c annual additions of members in the tabular proportions between the age
X and X'\'n.
Y Y
(39.) c '^^ V. — •^"^"'''' =the number of persons to which a Society joined by c persons
of the tabular ages x and under x-^-m would amount in n years. When a?+^>«^> this
formula will be reduced to the same form as equation (38.). And when x-^-m., as well as
^+^>ft/5 the equation becomes the same as (36.).
MDCCCLIX. 5 X
862 BIL FAEE 0^ THE COJSrSTEUCTION OP LIEE-TABLES
yil. LIEE-TABLE OF THE SIXTY-THEEE HEALTHIEST ENGLISH DISTEICTS.
Upon inquiry it was found that in many districts of England the mortality of the
population did not exceed the rate of 17 annual deaths to 1000 living.
For the sake of convenience these were called healthy districts, consisting of sixty-four,
or nearly a tenth part of the total registration districts of England and Wales, and inha-
bited by nearly a million of people : sixty-three of these districts have been taken as the
basis of the new Life-Table, constructed according to the methods previously described.
It will be seen that these districts, generally conterminous with Poor Law Unions, are
distributed over the various parts of the country. They comprise — Hendon (with Har-
row*) (17), Lewishmn (17), and JBromley (17) in the neighbourhood of London; Ham-
hledon (16), JDorJcing (17), Beigate (16), and Godstone (17) on the southern slope of the
Surrey hills; East Ashford (17) in East Kent, Blean (including Heme Bay) (17) be-
tween Canterbury and the sea; ten districts of Sussex — Battle (16) near Hastings, East-
hoiirne around Beachy Head (15), HailsJiam (17), UcJcJield (17), East Grinstead (17), Cuck-
field (16), Steyning near Brighton (16), Petworth (17), Worthing (17), and Midhiirst (17);
seven districts of Hampshire — the Isle of Wight separated from the mainland by the
sea (17), Lymington (17), Christchurch (16)^ Bingwood (17), Wew Forest (17), Gathering-
ton (17), and Alresford (17); Wokingham (17), and Easthampstead (16) in Berkshire,
south of the Thames; Ongar (17) in Essex, east of Epping Forest; Mutford (17), in-
cluding Lowestoft on the SuflPolk coast; Henstead (17), south of Norwich; Kingshridge
(17), on the south coast of Devon; Okehamjpton (16); Crediton (17), Barnstaple (17),
Torrington (17), Bideford (17), Holsworthy (16), stretching from the centre over Dart-
mouth and Exmoor, along the coast of the Bristol Channel; Stratton (17), Camelford
(17), and Laimceston (17), in the adjacent parts of Cornwall, and further south St. Co-
lumh (17); Williton (17) in Somerset, also on the Bristol Channel; Winchcomb (17), to
the east of Cheltenham, and the Cotswold Hills around the sources of the Thames ;
Kings Norton (17) in Worcestershire, adjoining Birmingham; Melton Mowbray (17) in
Leicestershire; Southwell (17) about Sherwood Forest, in the centre of Nottinghamshire ;
Garstang (16) in Lancashire, looking northward over Lancaster Bay; Easingwold (17)
in the North Hiding of Yorkshire, Guisborough (16) on the eastern coast north of Whitby ;
then follow five border districts of Northumberland on the southern face of the Cheviot
Hills: — Belford (17), Glendale (15), Bothbury (15), Bellingham (17), Haltwhistle (16)
(is omitted in the Table); Longtown (17) and Brampton (17) on the border, and Bootle
(16) on the coast of Cumberland, the East Ward (17) of Westmoreland, Haverfordwest
(17), on the western point of South Wales; Bnilth (16), Corwen (17), Pwllheli (17) on
Carnarvon Bay, and Anglesey (17) complete the list. These districts, and others nearly
equally healthy, have been thus described : —
" Such is the variety of the soil of England, that tested by the rates ot mortality, the
children reared out of a given nutaber born, the longevity of the inhabitants, the free-
* The annual deaths to 1000 living of all ages inserted in parentheses are deduced from returns of the
living at the censuses 1841 and 1851, and the deaths registered in the ten years 1841 to 1850. See
Eegistrar- General's Sixteenth Beport, pp. 141-153.
BE. TAEE ON THE CONSTEUCTION OF LIFE-TABLES. 863
dom from common epidemics, or the immunity froift cholera, Healthy Districts are found
in nearly every county. Large tracts of country are, however, so much healthier than
the rest, that they may be justly called Salubrious Fields ; and it is remarkable that here
the finest races of animals are bred. The north districts of Northumberland around the
beautiful CheviotHills, covered with grasses, ferns, wild thyme, — extending from the region
of the heaths to the rich cultivated land at their bases, touching each other, or intersected
by narrow valleys ; the districts extending from the Tees over the North and East Eidings
of York to Leicestershire, Herefordshire, and parts of Shropshire ; some of the districts of
Gloucestershire about the Cotswold Hills ; parts of "Wales ; North Devon, including Dart-
moor and Exmoor ; the Surrey and Sussex hills with the Southdowns, — have given names
to the best breeds of sheep, fowls, cattle, and horses in the kingdom." -^ * * ^ * ^
" The dry and most inland are not always the healthiest regions of the country. The
salubrious fields are sometimes watered by running streams, and diversified by lakes ;
the dew is abundant ; they are often veiled, not by infectious fogs, but by mists drawn
from the sky as it breathes over them ; the mountains rise above, the ocean rolls at the
distance below them, as on the coast of Sussex, North Devon, the western region of
Wales, extending under Snowdon and Cader Idris in a vast amphitheatre round Car-
digan Bay ; the lake land and moors of the North, rising between the Irish Sea and
the German Ocean. The land is sometimes heathy, but may be covered by the sweetest
herbage and bees feeding on the flowers : the cereal grains, the hop, the timber, are often
of the finest quality ; the animals are healthy, the native breeds are vigorous, and those
fine varieties are produced at intervals, which men of the genius of Bakewell, Ellman,
ToMKiNS, CoLLma, and O'Kelly make the permanent stock of the country. Industry
and the army receive their best recruits from the population ; while they get their worst
from the people of the low parts of sickly towns. Agriculture has reclaimed many
unhealthy districts on the plains, so that a considerable extent of the cultivated land is
now in a state of comparative salubrity; and vast systems of drainage have subdued the
noxious fens, although carried out less efficiently than is desirable, and interfered with
by milldams on the rivers, descending like the Nene from the inland high lands*."
The sanitary condition of the people in these districts is, however, still in many
respects defective.
CONCLUSION.
Halley first pointed out the financial applications of the Life-Table, and first cal-
culated the values of life annuities. That branch of science, in the various forms of
life insurance, has since received great developments. The new Table shows that the
duration of life, among large classes of the population, by no means in unexceptionable
sanitary conditions, exceeds the term of the ordinary Tables, and proves that life annui-
ties cannot be sold advantageously by offices, or by the Government, to large classes of
lives for less than the values deducible from the new Table.
A new branch of science has been developed since Halley's day, — it is the science of
Public Health. And here a new application of the Life-Table is found.
* Eeport to the Eegistrar- General on Cholera, pp. xcv, xevi.
5x2
864
BE. FAEE ON THE CONSTEUCTION OF LIFE-TABLES.
It is probable, upon physiological ^grounds, that man goes through all the phases of
his natural development in a hundred years ; and that the period of active life seldom
extends beyond eighty years. But this is a very indefinite measure, as the rates of
mortality, in all the intermediate ages, are left undetermined after it has been ascertained
in what proportions men attain the extreme limits.
Generations of men, under all circumstances, die at all ages; but the proportions
vary indefinitely under different conditions from a slight tribute to death each year,
dovfn to the point of extermination by pestilence. If we ascertain at what rate a gene-
ration of men dies away under the least unfavourable existing circumstances, we obtain
a standard by which the loss of life, under other circumstances, is measured ; and this I
have endeavoured to determine in the Life-Table of English Healthy Districts. And
recollecting that the science of public health was almost inaugurated in England by a
former President of this Society^, who encouraged and crowned the sanitary discoveries
of Captain Cook, I feel assured that it will receive with favour this imperfect attempt
to supply sanitary inquirers with a scientific instrument.
In a subsequent paper I hope to be able to lay before the Society the mortality by
different kinds of diseases at each age, as they have been deduced from the same series
of observations «
H
Table A. — Population, 1851. Deaths in the five years 1849 to 1853. Average Annual
Mortality per cent., and Logarithms of the Mortality.
Ag
es.
Population.
Deaths
•
Average annual
to 100 livin
mortality
g(w).
Logarithms of the mortality {^m).
Persons,
Bfales.
Females.
Persons.
Males.
Females.
Persons.
Males.
Females.
Persons.
Males.
Females.
I.
2.
996773
3-
4.
5-
6.
7*
8.
9-
10.
11.
12.
13-
All ages . .
493525
503248
87345
43736
43609
1*753
1-772
1*733
2*2436718
2-2485599
2*2388240
Under 5 . .
130635
65700
64935
26361
14282
12079
4*036
4-348
3*720
2-6059323
2*6382536
2-5705821
5— ••••
122406
61733
60673
4209
2080
2129
•688
•674
•702
3-8374062
3*8285759
3-8462102
10 — .
IIO412
56651
53761
2377
1087
1290
•431
'sH
•480
3-6340429
rsHosi^
3-6811523
IS
181339
90066
91273
6603
3113
3490
•728
•691
•765
3-8622801
3-8396482
3-8835130
zs— .
136892
65422
71470
5869
2675
3194
•857
•818
•894
3-9332160
3-9126300
3-9512411
35— •
108056
52734
55322
5208
2447
2761
•964
-928
-998
3-9840521
3-9675733
3-9991985
45
85244
42383
42861
5252
2698
2554
1-232
1-273
1-192
2*0906909
2-1048802
2-0761886
55
62857
31105
31752
7001
3568
3433
2*228
2-294
2-162
2*3478365
2*3606246
2-3349327
65- ..
39453
18860
20593
10313
5173
5 HO
5-228
5-486
4-992
2-7183350
2-7392308
2-6982734
75
16737
7718
9019
10297
4946
5351
12-304
12-817
11-866
1*0900631
1-1077793
1*0743066
85- .
2614
1097
1517
3581
1555
2026
27-399
28-350
26-711
1-4377287
1*4525536
7*4266838
95 and upwards
128
56
72
274
112
162
42-813 40-000
45-000
1-6315706
1-6020600
1*6532125
Wote, — Tke ages at death of 146 persons, viz. 123 males and 23 females, were not stated ; in calculating
the mortality they have been distributed proportionally over the several ages in the Table. The Table may
be read thus : 136,892 persons, of whom 66,422 were males, 71,470 were females at the age of 25 and under
35, were enumerated in 1851 ; at the same ages, 5869, 2675 males and 3194 females, died in the Rre years
1849 to 1853 ; consequently the annual rates of mortality per cent, were '857, '818, and '894.
* Sir JOHK PllINGLE.
DE. FAER ON THE C0N8TEUCTI0N OP LIFE-TABLES.
66
Number of Deaths at five
periods of Age in the
Healthy Districts, in
1848 to 1855.
Years.
Ages.
Persons.
Males.
Females.
0.
1.
2.
3.
4.
0.
1.
2.
3.
4.
0.
1.
2.
3.
4.
1848.
2935
832
458
371
312
1678
442
244
204
162
1257
390
214
167
150
1849.
2932
858
541
4.27
292
1637
452
263
207
154
1295
406
278
220
138
1850.
2969
859
466
331
301
1676
453
231
164
144
1293
406
235
167
157
1851.
3185
932
543
341
288
1769
502
274
179
148
1416
430
269
162
140
1862.
3405
860
567
389
297
1913
446
273
206
140
1492
414
294
183
157
1853.
3370
946
554
376
287
1888
514
293
179
137
1482
432
261
197
150
1864.
3404
1047
601
386
311
1903
539
317
197
165
1501
508
284
189
146
1855.
3350
907
533
445
297
1948
483
257
230
156
1402
424
276
215
141
Number of Births in Sixty-three Healthy Districts of England, 1848 to 1855,
1
2
3
4
Years.
Persons.
Males.
Females.
1848
28679
14756
13923
1849
29128
14751
14377
1850
29699
16176
14523
1851
30163
15465
14698
1852
30370
15557
14813
1853
29214
15010
14204
Males.
2) 29,507 = births in
births on
14,754:
13,117:
12,664:
12,390;
12,184:
12,047:
: living on
living on
diving on
:living on
: living on
1848 and
January 1,
January 1,
January 1,
January 1,
January I,
January 1,
1849
1849
1850
1851
1852
1853
1854
Age.
Males.
1637 = deaths in 1849
1
453 — deaths in 1850
2
274=deaths in 1851
3
206=:deaths in 1852
4
137 = deaths in 1853
866
DR. FAEE ON THE COKSTEUCTION OF LIFE-TABLES.
Table B.
-The several values of Kp^ on which the Life-Table of Healthy Districts is
based : also the corresponding values of j^^ and (1 —p^).
Age
= logarithms of the probability of living
one year after the age £c.
Px
s= probability of living a year.
(1 ~P^)
—probability of dying in a year.
Males.
Females.
Males.
Females.
Males.
Females.
1
2
3
7
12
20
30
40
50
60
70
80
90
r^9480215
1-9844929
r-9904341
T-9932422
r^9970729
P9984539
^9969724
P9964260
1^9959051
1-9943048
1-9895894
^9751357
r9420680
P8747315
T-9577796
T^9859276
T-9904679
1-9932928
r-9969512
1^9980197
1^9966528
1^9960967
T-9956263
P9946669
T^9902049
I^9773538
^9463182
P8809176
•88720
•96492
•97821
•98456
•99328
•99645
•99305
•99180
•99062
-98697
•97631
•94436
•87512
•74943
•90736
•96812
•97829
•98467
•99300
•99545
•99232
•99105
•98998
•98780
•97770
•94919
•88373
•76018
•11280
•03508
•02179
•01544
•00672
•00355
•00695
•00820
•00938
•01303
•02369
•05564
•12488
•25057
•09264
•03188
•02171
•01533
•00700
•00455
•00768
•00895
•01002
•01220
•02230
•05081
•11627
•23982
iVofe. — Age X is in this Table the precise age. Age 12 is applied frequently
to all persons of the age of 12 and under the age of 13; but in this Table it applies
only to persons of the precise age of 12 years, neither more nor less. The Xp^ was in
The >lPi29 deduced from this formula, is
both cases derived from the formula
o
2-\-m
for males 1*9983497, and for females 1*9979153; vrhich may be regarded either as
the constant or the mean values of Xp^^^ Xpu, Xp^^^ Xp^^^ and Xp^^ ; but as these are the
terminations of an ascending and a descending series, it is probable, and quite in con-
formity with other observations, that one, two, or more of these values will exceed the
mean value. The logarithms ofp^^ adopted are given above ; and the two arithmetical
means of the five logarithms, Xp^^^, Xp^^, Xp^^, Xp^^, and Xp^^^, resulting from the interpola-
tion, are 1*9983688 for males, and 1*9979435 for females.
The values of Xp^^^ Xp^^ . , . . are derived from the formula y^=10 ^^ ^^-'""^
BE. PAEE ON THE CONSTEUCTION OE LIFE-TABLES.
867
NOTE ON THE TWO HYPOTHESES.
Let h be the decrement of the ordinate ^ in a
unit of time, then the decrement Ay of the ordi-
nate in the time x^ represented by the abscissa,
will be Ay=—bw, on Demoiyre's hypothesis;
and as it is always proportional to the time, it
will be in an infinitely short time dy= — bdx.
Passing to the integral yzuc—hx. And ity=a
at the origin when ^==0, c=a^ .*. y=-a'--hx. And
if J = 1 , then y=,a—x. This evidently represents
very closely short portions of the Life-Table
curve ; and the smaller x is taken, the nearer is the
approximation to the corresponding value of y.
Again, let Ay be the decrement of the ordinate
y in the indefinite time A^ represented by the
abscissa; and let the mortality (m) represented
by the ratio of the area dbfg to the area dfg be
Numbers ^—
living. 3Q
70,000
Ages.
31
32
60,000
60,000
=mo. Let also m^ increase at the rate r in a 40,000
unit of time, so that T--7=j^=mi=mor, and
generally within given limits m^T'-=>m^\ then
Ay=— ^m^A^ nearly, A^ being any small por-
tion of time. 30,000
The error increases as the time A^ is extended,
from the circumstance that on the one hand m^
varies by hypothesis momentarily, and that y,
from which the varying proportional part is taken,
constantly grows shorter. But by passing to the 20,000
limit and making the time dx infinitely short,
m^ and y during that infinitely short time may
be considered constant, and 6?^/= — ym^dx will be
the true decrement. Substituting rnQf for m^,
the equation becomes dy= —ym^r'dx^ from which ^^'^^
the value of y can be derived, as before shown.
For — =: — mor^'c?^, and integrating both sides
y
\y=\e
Here X, stands for the logarithm
having s for its base.
At the origin of the curve, when ^==0, let 3/ = !, and then XsC
40
41
42
I
d
f
c
n
a
Air*
Now substituting
868 BE. EAEE ON THE CONSTETJCTION OF LIFE-TABLES.
this value for \€, we have X,y=-^ — ^, .-. X,t/='~(l—r'); and passing to the
t 6
number, y=:g^^^~ . Putting k for the modulus of the common logarithm (X) having 10
for its base, we have X,y=-j, and K,r=-j, ,\ ~^="^ (1— r") ; or passing to the number,
Upon the one hypothesis, out of a generation of men an equal quantity of life^ is
destroyed in equal times, out of diminishing quantities in existence, the proportion that
perishes of the residual life constantly increasing.
Upon the other hypothesis, a decreasing proportion of the residual life is destroyed
from birth down to the age of puberty ; in the after ages, a proportion increasing at
diiferent rates is destroyed in equal times. The quantity of life destroyed in equal times
may be the same, or different upon this hypothesis. And in very short intervals of age
the differences between the quantities of life destroyed may be so inconsiderable, that
they may be neglected.
The two hypotheses may be illustrated. Assume that at every beat of the heart an
equal quantity of vital force on an average is consumed in excess of that produced ; or
if this does not happen at distant ages, assume that it happens during two consecutive
years, two consecutive days, two consecutive pulses of a generation of men, and is repre-
sented by the deaths in the two intervals ; this will give an idea of the first hypothesis.
The second hypothesis will be represented by assuming that, in addition to the exist-
ing force, a certain amount of vital force is produced, while a certain amount is also
destroyed at every beat of the heart ; the quantity destroyed exceeding the quantity
produced in a diminishing ratio, and then in an increasing ratio ; the proportional part
destroyed being for this purpose always represented by the proportional number of
hearts beating to the number of hearts ceasing to beat at every instant of age, among a
generation of men. The respirations, the sensations, the secretions, nutrition, and all
the vital acts may be conceived like the heart to influence the continuance of the vital
force ; implying here simply the force which sustains life.
* The quality or the intensity of life at different ages is purposely left out of consideration,
June 16, 1859.
DE. FAEE ON THE CONSTETJCTION OP LIFE-TABLES.
869
Table B1.— LIFE-TABLE OF HEALTHY ENGLISH DISTRICTS.
Logarithms of the Numbers of Males and Females living at each year of age.
X^
1
'X'
X'#.
Age.
X,
Males.
Age.
X.
Females.
Age.
X.
Males.
Age.
X,
Females.
4.7086364
1
4.6890835
56
4.4361998
56
4-4177773
1
4.6666579
1
4.6468631
56
4-4279544
56
4.4116015
2
4.6411608
2
4-6327907
57
4.4203212
57
4.4062190
3
4.6316849
3
4»6232586
m
4.4122719
58
4.3981622
4
4.6248271
4
4.6166614
5g
4.4037768
59
4.3901691
5
4.6193109
6
4.6110606
60
4-3943905
60
4-3812819
6
4.6148376
6
4-6066737
61
4-3839799
61
4-3714868
7
4.6112225
7
4.6028960
62
4-3726154
62
4.3607637
8
4.6082964
8
4.5998462
63
4.3699618
63
4-3490765
9
4.6069001
9
4.6972668
64
4.3462281
64
4-3363727
10
4.6038946
10
4.6960094
65
4.3312678
65
4.3226837
11
4.6021611
11
4-5929497
66
4-3149786
66
4.3076249
12
4.6006660
12
4.5909763
67
4.2972628
67
4.2913961
n
4.59901 00
13
4.6889960
68
4.2779668
68
4-2737774
14
4-5974279
14
4.5869326
69
4.2669814
69
4.2646384
i5
4.5967387
i5
4.5847269
70
4.2341418
70
4-2338287
16
4-5938865
16
4.5823368
71
4.2092775
71
4.2111825
17
4.6918269
17
4-5797373
72
4.1822024
72
4-1866180
18
4.6896314
1,8
4.5769202
73
4.1627146
73
4.1696372
19
4-6869878
19
4-5738947
74
4.1206968
74
4.1303269
20
4.6841961
20
4-6706868
75
4.0856167
75
4-0983537
21
4.6811675
21
4.5673396
76
4.0476228
76
4.0634741
22
4.6780627
22
4-6639166
77
4-0060634
77
4-0264242
■ 23
4.6748607
23
4.5604237
78
3.9609277
78
3-9839262
24
4.5716008
24
4.5568665 ■
79
3-9118498
79
3.9386819
25
4.5682808
26
4.5532498
80
3-8686083
80
3-8893831
26
4.6649078
26
4-5495779
81
3.8006763
81
3-8367013
27
4.5614874
27
4.5458646
82
3-7377111
82
3.7772929
z%
4.6680244
28
4.5420830
83
3.6696642
83
3-7137979
29
4.5546223
29
4.6382666
84
3-5967318
84
3-6448405
30
4.5609836
30
4-5344046
85
3.5168641
85
3.5700284
31
4.5474095
31
4.5306013
86
3.4296159
86
3-4889532
32
4«5438oo5
32
4.5265566
87
3-3362962
87
3.4011904
33
4.5401667
33
4.5226708
88
3-2357683
88
3.3062992
34
4.5364730
34
4.5186435
89
3.1274600
89
3.2038228
35
4.5327494
35
4.5144739
90
3.0109034
90
3-0932880
36
4.5289808
36
4.5103606
91
2.8866349
91
2.9742066
37
4.5261620
37
4.5062016
92
2.7611463
92
2.8460701
38
4.5212864
38
4.6019942
93
2.6069196
93
2-7083699
39
4.5173467
39
4-4977353
94
2.4624273
94
2.5606372
40
4.5133342
40
4-4934212
95
2.2871223
95
2.4020479
41
4.5092393
41
4.4890475
96
2.1104426
96
2.2323219
42
4.6060612
42
4.4846093
97
1.9218108
97
2.0607729
43
4.5007679
43
4.4801012
98
1.7206337
98
1-8667982
44
4.4963465
4z,
4.4766172
99
1.5063024
99
1-6497793
45
4.4918029
45
4.4708606
100
1*2781926
100
1.4290811
46
4.4871119
46
4.4660943
101
1.0366640
101
1.1940626
47
4.4822670
47
4.4612404
102
0.7780608
102
0.9440265
48
4.4772210
48
4.4662807
103
0.5047118
103
0-6783194
49
4.4719862
49
4.4612061
104
0.2149296
104
0.3962318
50
4.4666301
50
4.4460074
io5
9-9080117
io5
0.0970476
51
4.4608349
51
4.4406743
106
9-6832396
106
9.7800361
52
4.4648778
52
4.4361962
107
9-2398792
107
9*4444460
53
4.4486368
53
4.4296620
108
8.8771808
io8
9-0895160
54
4.4420848
54
4.4237698
109
8-4943792
109
8.7144646
The above Tables were calculated and stereogiyphed by Scheutz's Calculating Machine at the General Begister Office, Somerset
House. The impression was made by the machine on jpajpier macM in the dry state. Sheet lead received the impressions in the
original invention. The use of papier macM was suggested by Mr. W. Mattkess, Overseer in the Firm of Messrs. Taylor and
Francis. In the wet state^ as it is used by stereotype founders, papicT macM did not however succeed ; but after several trials, it
was found that dry papier macM^ black-leaded, supplies a good mould for the stereotype metal.
MDCCCI.IX. 5 Y
870
BE. EAEE ON THE CONSTEUCTION OF LIFE-TABLES.
m
Q
r 1
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871
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872
DE. FAEE 01^ THE CONSTEUOTION OF LIFE-TABLES.
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873
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875
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MDCCCLIX.
5 z
878
DE. EAEE OJSr THE COlSrSTEirCTIOlSr OE LIFE-TABLES.
Table G.— HEALTHY DISTEICTS LIFE-TABLE.
The Mean Afteb-lifitime (or the Hw^pectation oflAfe) at the age ^5 and at the age w
and upwards ; also the Meaf Ages of the Livme and the Mean Ages at Death.
(Constructed from Tables D, E, F.)
]
PERSONS.
'
Mean Age
\ at Death
Age
(or past
Lifetime).
Mean Afte^-
Mean After-
Mean Age of
lifetime of
Persons of the
Age <«*.
lifetime of
Persons of the
Age 35 and upwards.
Persons living
of the Age x
and upwards.
Of Persons
actually living
at the Age oo.
Of Persons
actually living
at the Age ^
and upwards.
A ^
X A^.
w^ka.
w^2k'^.
49*00
33*92
33*9^
49 'OO
67*84
5
54'i6
31*9^
36-98
59*i6
68*96
10
51*08
29*91
39*91
6ro8
69*82
15
47*12
27*85
42*85
62*12
70*70
20
43*45
25*82
45-82
63*45
71*64
^5
40*05
23*79
48*79
65*05
72*58
30
36*64
21*76
51*76
66*64
73*52
35
33*17
19*73
54*73
68*17
74*46
40
29*64
17*71
57*71
69*64
75*42
45 .
26*05
15*71
60*71
71*05
76-42
50
22*44
13*74
6374
72*44
77-48
55
i8*86
11*84
66*84
73*86
78*68
60
15*37
10*04
70*04
75*37
8o*o8
65
12*29
8*37
73*37
77*29
81*74
70
9-61
6*86
76*86
79*61
83*72
75
7*34
5*52
80*51
82*34
86*02
80
5*51
4*36
84*36
85*51
88*72
85
4*10
3*41
88*41
89*10
91*82
90
3*05
2*65
92*65
93*05
95*30
.95
2*29
2*05
97-05
97*29
99*io
100
1*72
1*47
101*47
101*72
102*94
Age
(or past-
Lifetime).
MALES.
FEMALES.
Mean After-lifetime
of Males of the Age op.
Mean Age at Death
of Males actually
living at the Age x.
Mean
After-lifetime of
Females of the Age oe.
Mean Age at
Death of Females
actually living
at the Age os.
w.
A ^^
05-\-kx*
A~^-
X-^kx*
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
48*56
54*39
51*28
47*20
43*40
39*93
36-45
32*90
29*29
25*65
22*03
18*49
15*06
12*00
9:37
7*15
5*37
4*01
2*99
2*25
1*69
48*56
59*39
61*28
62*20
63*40
64*93
66*45
67*90
69*29
70*65
72*03
73*49
75-06
77*00
79*37
82*15
85*37
89*01
92*99
97*25
101*69
49*45
53*93
50*88
47*04
43*50
40*18
36*85
33*46
30*00
26*46
22*87
19*24
15*69
12*58
9*85
7*52
5*64
4*19
3*11
2*32
1*75
49*45
58*93
60*88
62*04
63*50
65*18
66*85
68*46
70*00
71*46
72*87
74*24
75-69
77*58
79*85
82*52
85*64
89*19
93*11
97*32
101*75
The Table may be read thus j— Persons in the Healthy Districts of England of the precise age 20 will live on an average 43*45 years ;
while persons of the age of 20 and upwards^ living in a normally constituted population of the same character, will live on an average
25*82 years. The mean age of persons of the age 20 and upwards is 45*82 years ; the mean age at death of persons living at the
precise age 20 will be 63*45, while the mean age at death of persons actually living at the age x and upwards will be 71*64 years.
JfwrWers Dying
21,000
10,000
9.
ooo
8,000
y.OOO
€,OO0
5,00
4-jOOO
3,000
2,000
IQOO
Year of Age o_.
Nwmhers ttving
200,000
(]0,
000
80,000 „
y 0,000
60,000
60, 000
40,000
30,000
2O,00O
20, a 00
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HEALTHY DISTRICTS.
LIFE TABLE DIAGRAMS .
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